let i be an interval in r, when is f said to be concave on i

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Colby Messinger answered 1 year ago

Let A ⊆ R, let f : A → R be a function, and let c...

Let A ⊆ R, let f : A → R be a function, and let c be a limit point of A. Suppose that a student copied down the following definition of the limit of f at c: “we say that limx→c f(x) = L provided that, for all ε > 0, there exists a δ ≥ 0 such that if 0 < |x − c| < δ and x ∈ A, then |f(x) − L| < ε”. What was...

Let R be a UFD and let F be a field of fractions for R. If...

Let R be a UFD and let F be a field of fractions for R. If f(α) = 0, where f ∈ R [x] is monic and α ∈ F, show that α ∈ R NOTE: A corollary is the fact that m ∈ Z and m is not an nth power in Z, then n√m is irrational.

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Let R be a commutative domain, and let I be a prime ideal of R. (i)...

Let R be a commutative domain, and let I be a prime ideal of R. (i) Show that S defined as R \ I (the complement of I in R) is multiplicatively closed. (ii) By (i), we can construct the ring R1 = S-1R, as in the course. Let D = R / I. Show that the ideal of R1 generated by I, that is, IR1, is maximal, and R1 / I1R is isomorphic to the field of fractions of...

2 Let F be a field and let R = F[x, y] be the ring of...

2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F. (a) Prove that ev(0,0) : F[x, y] → F p(x, y) → p(0, 0) is a surjective ring homomorphism. (b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]} (c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y]...

6. (a) let f : R → R be a function defined by f(x) = x...

6. (a) let f : R → R be a function defined by f(x) = x + 4 if x ≤ 1 ax + b if 1 < x ≤ 3 3x x 8 if x > 3 Find the values of a and b that makes f(x) continuous on R. [10 marks] (b) Find the derivative of f(x) = tann 1 1 ∞x 1 + x . [15 marks] (c) Find f 0 (x) using logarithmic differentiation, where f(x)...

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f...

5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is injective, but not surjective. (b) Suppose g : R\{−1} → R\{a} is a function such that g(x) = x−1, where a ∈ R. Determine x+1 a, show that g is bijective and determine its inverse function.

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R. (i)...

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R. (i) List explicitly all the ideals A of R with I ⊆ A. (ii) Write out all the elements of R/I (these are cosets). (iii) List explicitly the set of all ideals B of R/I (these are sets of cosets). (iv) Let π: R → R/I be the natural projection. For each ideal A of R such that I ⊆ A, write out π(A) explicitly...

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f...

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f rises strictly monotonously ⇒ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?) If f is reversable, f has no critical point. (TRUE or FALSE?) If a is a “minimizer” of f, then f ′(a)...

Let f: A → B be a function, and let {Bi: i ∈ I} be a...

Let f: A → B be a function, and let {Bi: i ∈ I} be a partition of B. Prove that {f−1(Bi): i ∈ I} is a partition of A. If ~I is the equivalence relation corresponding to the partition of B, describe the equivalence relation corresponding to the partition of A. [REMARK: For any C ⊆ B, f−1C) = {x ∈ A: f(x) ∈ C}.]

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